Asymptotic Properties of Backfitting Estimators

By Jean D. Opsomer. 2000

171 citations recorded by the source corpus.

When additive models with more than two covariates are fitted with the backfitting algorithm proposed by Buja et al. [2], the lack of explicit expressions for the estimators makes study of their theoretical properties cumbersome. Recursion provides a convenient way to extend existing theoretical results for bivariate additive models to models of arbitrary dimension. In the case of local polynomial regression smoothers, recursive asymptotic bias and variance expressions for the backfitting estimators are derived. The estimators are shown to achieve the same rate of convergence as those of univariate local polynomial regression. In the case of independence between the covariates, non-recursive bias and variance expressions, as well as the asymptotically optimal values for the bandwidth parameters, are provided.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/87887265551b7033070606afffd938feb13db2e5

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In Statistics and Probability

Citations in the corpus

Smooth backfitting in generalized additive models (2008, background): “Opsomer and Ruppert [22] and Opsomer [21] showed that the backﬁtting estimator is well-deﬁned asymptotically when the stochastic dependence between covariates is “not far” from independence.”
Nonparametric inference with generalized likelihood ratio tests (2007, background): “…h1 = 2 3h1,opt, h1,opt, or 3 2h1,opt with h2 fixed at its optimal value h2,opt, where hd,opt is the optimal bandwidth for the smoother on md(·) (see Opsomer, 2000); and (ii) three levels of nuisance function m1(X1): m1,β(X1) = [ 1 + β √ var(0.5− 6X21 + 3X31 ) ] (0.5− 6X21 + 3X31 ), where β =…”
Some theory for penalized spline generalized additive models (2002, methodology): “Equation (7) is comparable to the result of Lemma 2.1 in Opsomer (1999) , where, for local polynomial estimators using backtting, a similar identity can be obtained.”
Nonparametric estimation of an additive model with a link function (2002, background): “Linton and Nielsen (1999); and Opsomer (2000) have investigated the properties of backfitting”
Time-Varying Additive Models for Longitudinal Data (2013, methodology): “With strict constraints on the dependence between covariates, Opsomer and Ruppert (1997) and Opsomer (2000) showed that the ordinary backfitting method works but cannot achieve the oracle bias.”
Local Likelihood Estimation in Generalized Additive Models (2003, methodology): “As for additive models, the bias and variance expressions become much more complicated for models with q > 2 (see Opsomer, 2000).”
FWDselect : Variable selection algorithm in regression models (2013, unknown): “We estimate the latter using the backfitting algorithm (Opsomer, 2000).”
Smooth backfitting in practice (2005, methodology): “The first issue is of interest because it is well known from theory (see Opsomer and Ruppert (1997) or Opsomer (2000) and Linton (1997)) and from practice (see Sperlich et al. (1999) or Dette et al.”
Optimal estimation in additive regression models (2006, background): “Opsomer (2000) considered backfitting estimators that use local polynomial estimators in the iterative updates of the additive components.”
Local Inference in Additive Models with Decorrelated Local Linear Estimator (2019, background): “In the low-dimensional setting, additive models have been carefully investigated from both the methodological and theoretical perspectives [4,20,21,25,27,41].”
Backfitting and smooth backfitting for additive quantile models (2010, methodology): “…backfitting procedure of Buja, Hastie and Tibshirani (1989), whose theoretical properties were studied later by Opsomer and Ruppert (1997) and Opsomer (2000), the marginal integration technique of Linton and Nielsen (1995), and the smooth backfitting of Mammen, Linton and Nielsen (1999),…”
On the use of generalized additive models in time-series studies of air pollution and health. (2002, background): “Asymptotic bias and variance properties of backfitting estimators have been explored by Opsomer and Rupper (27) and Opsomer (28).”
Efficient and fast spline-backfitted kernel smoothing of additive models (2009, background): “Opsomer (2000) extended the theoretical results to a general case with more than two covariates.”